Position Vector

4 Key excerpts on "Position Vector"

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  eBook - PDF

  Introductory Mathematics for Engineering Applications

  • Kuldip S. Rattan, Nathan W. Klingbeil, Craig M. Baudendistel(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  Two-Dimensional Vectors in Engineering CHAPTER 4 The applications of two-dimensional vectors in engineering are introduced in this chapter. Vectors play a very important role in engineering. The quantities such as displacement (position), velocity, acceleration, forces, electric and magnetic fields, and momentum have not only a magnitude but also a direction associated with them. To describe the displacement of an object from its initial point, both the distance and direction are needed. A vector is a convenient way to represent both magnitude and direction and can be described in either a Cartesian or a polar coordinate system (rectangular or polar forms). For example, an automobile traveling north at 65 mph can be represented by a two-dimensional vector in polar coordinates with a magnitude (speed) of 65 mph and a direction along the positive y-axis. It can also be represented by a vector in Cartesian coordinates with an x-component of zero and a y-component of 65 mph. The tip of the one-link and two-link planar robots introduced in Chapter 3 will be represented in this chapter using vectors both in Cartesian and polar coordinates. The concepts of unit vectors, magnitude, and direction of a vector will be introduced. 4.1 INTRODUCTION Graphically, a vector −− → OP or simply  P with the initial point O and the final point P can be drawn as shown in Fig. 4.1. The magnitude of the vector is the distance between points O and P (magnitude = P) and the direction is given by the direction of the y Magnitude = P P x O θ Figure 4.1 A representation of a vector. 107 108 Chapter 4 Two-Dimensional Vectors in Engineering arrow or the angle  in the counterclockwise direction from the positive x-axis as shown in Fig. 4.1. The arrow above P indicates that P is a vector. In many engineering books, the vectors are also written as a boldface P.

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  eBook - PDF

  Electromagnetics and Transmission Lines

  Essentials for Electrical Engineering

  • Robert Alan Strangeway, Steven Sean Holland, James Elwood Richie(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  1 Vectors, Vector Algebra, and Coordinate Systems CHAPTER MENU 1.1 Vectors, 1 1.2 Vector Algebra, 4 1.2.1 Dot Product, 4 1.2.2 Cross Product, 7 1.3 Field Vectors, 10 1.4 Cylindrical Coordinate System, Vectors, and Conversions, 12 1.4.1 Cartesian (Rectangular) Coordinate System: Review, 12 1.4.2 Cylindrical Coordinate System, 13 1.5 Spherical Coordinate System, Vectors, and Conversions, 19 1.6 Summary of Coordinate Systems and Vectors, 25 1.7 Homework, 27 Motivation Why spend a chapter on vectors? The degree to which you master the vector concepts and techniques in this chapter will generally determine the degree to which you understand and can apply the essential concepts and techniques of electromagnetic fields. Vectors are not just important tools in the calculation of electromagnetic field quantities. They are essential in the visualization of electromagnetic fields in an organized, dependable man- ner. In short, a mastery of vectors instills the “thought infrastructure” that allows one to visualize and apply elec- tromagnetic fields in electrical engineering applications. Enjoy! 1.1 Vectors This chapter develops the tools necessary to define and manipulate various types of vectors in three different coor- dinate systems. We begin with how vectors can be used to describe location and displacement. What is a vector? It is a quantity with magnitude (scalar) and direction. What are examples of vectors? Velocity, force, acceleration, and so forth. How is direction in three-dimensions expressed? Start with the following vector definition: A Position Vector r locates a position in space with respect to the origin, that is, it is a vector that starts at the origin and ends at the point of the designated position in space (notation r is also used in some textbooks and literature). 1 Electromagnetics and Transmission Lines: Essentials for Electrical Engineering, Second Edition. Robert A. Strangeway, Steven S. Holland, and James E.

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  eBook - PDF

  Vector Calculus

  • William Cox(Author)
  • 1998(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  To get an insight into the new view-point, let us stick with our two-dimensional projectile. You may have a number of different definitions, but they are most likely to come from the following list: 1. Any quantity having both a magnitude and a direction, such as velocity as opposed to speed. 2. An n-tuple of real numbers viewed as a member of an n-dimensional Euclidean space. 3. An element of a vector space. Definition 1 is the one we meet at the most elementary level. Certainly, our position Functions of Q Vector 137 vector ret) seems to fit this description, if we think of it in the traditional way as a 'directed line segment' -an 'arrow'. But there is more to it than that. The represen-tation as a vector ret) does more than replace x, y by a single vector -it 'hides' the coordinate system altogether. If we choose a different coordinate system -say, by rotating the x-and y-axes, as we might do if considering motion on an inclined plane -then this would only change the representation ofr(t) in terms of the coordi-nate system; it would not change ret) itself. Consider a rotation about the origin of the x-and y-axes through an angle () in an anti-clockwise sense to a new set of axes x', y'. Express the coordinates (x', y') of a point P referred to the x' -and y' -axes in terms of the coordinates (x, y) referred to the x-and y-axes. ~ Ifwe regard (x, y) as the components of a vector r = OPrelative to the x-and y-axes and (x', y') as the components of the same vector relative to the x' -, y' -axes, then it is a standard exercise in coordinate geometry to show that under the rotation of axes the components ofr transform according to x' = (cos (})x + (sin (})y y' = -(sin (})x + (cos (})y Any quantity whose components transform in this way is called a two-dimen-sional vector. This is the key to the general definition of a vector -its behaviour under a particular transformation of the coordinate system.

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  Learning and Teaching Mathematics using Simulations

  Plus 2000 Examples from Physics

  • Dieter Röss(Author)
  • 2011(Publication Date)
  • De Gruyter

    (Publisher)

  In quantum mechanics one works with vectors in the infinitely dimensional Hilbert space. Plane problems can be described by two-dimensional vectors that can be considered to lie in the complex plane. Vector algebra and vector analysis, in which partial differentiations take place, are an especially important mathematical tool of theoretical physics and therefore are often treated in depth in many textbooks for first year students.Their objects and oper-ations are not easily accessible to the untrained imagination. Therefore, the following sections concentrate only on the interactive visualization of fundamental aspects. 8.2 3D-visualization of vectors The classical visual presentation of a vector is an arrow in space, whose length defines an absolute value and whose orientation defines a direction. The place at which the arrow is situated is arbitrary; one can, for example, let it start as a zero-point vector from the origin of a Cartesian system of coordinates. Thus its endpoint (the tip of the arrow) is described by the three space coordinates x; y; z in this system of coordinates. Its length a , also referred to as the absolute value of the vector, is obtained from the theorem of Pythagoras as a D p x 2 C y 2 C z 2 . It obviously does not matter how the system of coordinates, with respect to which the coordinates of the vector are defined, is orientated in space. Under a change of the coordinate system (translation or rotation), the individual coordinates also change, but the position and length of the vector are not affected by this. They are invariant under translation and rotation. This property provides the definition of a vector. Quantities that can be characterized by specifying a single number for every point in space are called scalar , in contract to vectors; an example would be a density-or temperature distribution. The three-dimensional zero-point vector represents the position coordinates of a point in space.

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